D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering

1. Planning operation start timesfor the manufacture of capital products with uncertain processing times and resource constraints D.P. Song, Dr. C.Hicks&Dr. C.F.Earl Department of MMM Engineering University of Newcastle upon Tyne ISAC, Newcastle upon Tyne, on 8-10 Sept., 2000.

2. Overview 1. Introduction 2. Problem formulation 3. Perturbation Analysis (PA) method 4. Simulated Annealing (SA) method 5. Case studies 6. Conclusions

3. Introduction -- a real example Number of operations = 113; Number of resources=13.

4. Introduction -- a simple example Final assembly Subassembly Component Component

5. Introduction -- operation start times • Si -- part or operation start times • Result in waiting times if {Si } is not well designed.

6. Introduction -- backward scheduling This seems perfect, but we may have uncertain processing time and finite resource capacity.

7. Distribution of processing time Distribution of completion time tardy probability Uncertainty results in a high probability of tardiness. Introduction -- uncertainty problem

8. Introduction -- resource problem Part 2 and part 3 use the same resourceÞ part 2 is delayed, part 1 is delayedÞ results in waiting times and tardiness.

9. Problem formulation • Find optimal S=(S1, S2, …, Sn) to minimise expected total cost: • J(S) = E{S(WIP holding costs • + product earliness costs • + product tardiness costs)} • Assumption: operation sequences are fixed. • Key step of Stochastic Approximation is: ¶J(S)/¶Si = ?

10. Perturbation analysis-- general problem • Consider to minimise: J(q) = EL(q,w) J(.)-- system performance index. L(.)-- sample performance function. q -- a vector of n real parameters. w -- a realization of the set of random sequences. • PA aims to find an unbiased estimator ofgradient -- ¶J(q)/¶qi, with as little computation as possible.

11. Calculate ¶L(q,w)/¶qi , i = 1, 2, …, n sample function gradient Perturbation analysis -- main idea • Based on a single sample realization • Using theoretical analysis • Exchange E and ¶: • ? E¶L(q,w)/¶qi= ¶EL(q,w)/¶qi • = ¶J(q)/¶qi

12. PA algorithm -- concepts • Two concepts: nominal path and perturbed path • Sample realization for {Si}-- nominal path (NP) • Sample realization for {Si+D, Sj,j¹i} -- perturbed path (PP), where D is sufficiently small. • All perturbed paths are theoretically constructed from NP rather than from new experiments

13. PA algorithm -- Perturbation rules • Perturbation generation rule -- When PP starts to deviate from NP ? • Perturbation propagation rule -- How the perturbation of one part affects the processing of other parts? • -- along the critical paths • -- along the critical resources • Perturbation disappearance rule -- When PP and NP overlaps again ?

14. perturbation generation perturbation disappearance PA algorithm -- Perturbation rules • If S2 is perturbed to be S2+ D. • Cost changes due to the perturbation.

15. perturbation generation perturbation propagation PA algorithm -- Perturbation rules • If S3 is perturbed to be S3+ D. • Cost changes due to the perturbation.

16. PA algorithm -- gradient estimate • From PP and NP calculate sample function gradient :¶L(S,w)/¶Si • -- usually can be expressed by indicator functions. • Unbiasedness of gradient estimator: • E¶L(S,w)/¶Si = ¶J(S)/¶Si • Condition: processing times are independent • continuous random variables.

17. Stochastic Approximation • Iteration equation: qk+1 = qk+1 + gk×ÑJk step sizegradient estimator of ÑJ • Combine PA and Stochastic Approximation => PASA algorithm to optimise operation start times

18. Simulated Annealing algorithm • Random local search method • Ability to approximate the global optimum • Outer loop -- cooling temperature (T) until T=0. • Inner loop -- perform Metropolis simulation with fixed T to find equilibrium state

19. Simulated Annealing algorithm • In our problem, a solution = (S1, S2, …, Sn). • A neighborhood of a solution can be obtained by making changes in Si. • New solution is adjusted to meet precedence and resource constraints; non-negative. • Cost is evaluated by averaging a set of sample processes.

20. Simulated Annealing algorithm

21. Example 1 -- multi-stage system • Product structure and resource constraints • Assume: Normal distributions for processing times. • There is noanalytical methods to solve this problem.

22. Convergence of cost in PASA J(S) Using Perturbation Analysis Stochastic Approximation to optimise operation start times.

23. Euclidean norm of gradient in PASA • Euclidean norm = ||Jn||

24. Compare PASA with Simulated Annealing Method J(S) SA1 23.94 SA2 23.93 SA3 23.92 SA4 24.11 ---------------- PASA 23.90 Where Simulated Annealing uses four different settings (initial temperature and number for check equilibrium) Compare the convergence of costs over CPU time (second).

25. Example 2-- complex system • Complex product structure with Normal distributions.

26. Resource constraints Resources Operation sequences 1000:247, 243, 239, 234, 231, 246, 242, 238, 230, 245, 237, 229, 228. 1211:236:1, 236:2, 236:3, 236:4, 236:5, 236:6, 236:7, 226:1, 236:8, 226:2, 226:3, 226:4, 226:5, 226:6, 236:11, 226:7, 232:1, 226:8, 235:1, 232:2, 236:12, 235:2, 226:9, 232:3, 235:3, 240:1, 235:4, 240:2, 226:10, 232:5, 236:13, 233:2, 235:5, 240:3, 233:3, 235:6, 240:4, 232:7, 226:11, 233:4, 235:7, 240:5, 232:8, 233:5, 235:8, 240:6, 232:9, 233:6, 240:7, 226:12, 232:10, 235:9, 240:8, 233:8, 240:9, 233:9, 226:13, 235:10, 240:10, 236:15, 226:14, 240:11, 236:16, 226:15. 1212:236:9, 236:10, 232:4, 232:6, 236:14, 232:11, 232:12. 1511:233:1, 233:7, 233:11.

27. Resource constraints Resources Operation sequences 1129:233:10. 1224:233:12. 1222:244:1, 244:3, 244:5, 241:1, 241:2, 241:3, 248:2, 248:3, 248:5, 248:6. 1113:244:2, 241:4, 241:5, 248:4. 1115:241:6, 241:7. 1315:244:4. 1226:244:6, 244:7. 1125:244:8, 248:7, 248:8. 1411:244:9, 248:1. Number of resources: 13. Total number of operations: 113.

28. Convergence of cost in PASA J(S) Using Perturbation Analysis Stochastic Approximation to optimise operation start times.

29. Euclidean norm of gradient in PASA ||Jn||

30. Compare PASA with Simulated Annealing Method J(S) SA1 121.74 SA2 124.60 SA3 121.78 SA4 124.90 ---------------- PASA 120.79 with four different settings Compare the convergence of costs over CPU time (minute).

31. Conclusions • Both PASA and SA can deal with complex systems beyond the ability of analytical methods. • PASA is much faster and yields better solutions than Simulated Annealing in case studies • SA is more robust and flexible, does not require any assumption on uncertainty

32. Further Work • Optimise both operation sequences and start times • Integrate Perturbation Analysis with SA or Evolution algorithms • Extend to dynamic planning problems such as incremental planning and re-planning